The free field realisation of the BVW string

The symmetric orbifold of 𝕋4 was recently shown to be exactly dual to string theory on AdS3 × S3 × 𝕋4 with minimal (k = 1) NS-NS flux. The worldsheet theory is best formulated in terms of the hybrid formalism of Berkovits, Vafa & Witten (BVW), in terms of which the AdS3× S3 factor is described by a psu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{psu} $$\end{document}(1, 1|2)k WZW model. At level k = 1, psu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{psu} $$\end{document}(1, 1 2)1 has a free field realisation that is obtained from that of u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{u} $$\end{document}(1, 1 2)1 upon setting a u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{u} $$\end{document}(1) field, often called Z, to zero. We show that the free field version of the N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 generators of BVW (whose cohomology defines the physical states) does not give rise to an N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 2 algebra, but is rather contaminated by terms proportional to the Z-field. We also show how to overcome this problem by introducing additional ghost fields that implement the quotienting by Z.


Introduction
The string theory background dual to the actual symmetric orbifold of T 4 has recently been identified in [1,2], see also [3,4] for earlier work. The string background has pure (and minimal, i.e. k = 1) NS-NS flux, and can therefore be described be an exactly solvable worldsheet WZW model. Given the familiar problems with the k = 1 theory in the usual R-NS approach of [5], the best description utilises the so-called hybrid formalism of Berkovits,

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Vafa and Witten (BVW) [6]. Then the relevant WZW model is based on the superalgebra psu(1, 1|2) k , for which the case k = 1 is unproblematic. From this viewpoint the breakdown of the naive R-NS description at k = 1 manifests itself in that psu(1, 1|2) k only possesses short representations at level k = 1 [1].
The full worldsheet theory of the BVW hybrid string consists of a psu(1, 1|2) k WZW model, together with a topologically twisted T 4 theory. This is to say, the four free bosons of T 4 are conventional free fields (and their derivatives define fields of conformal dimension h = 1), while of the four free fermions, two have conformal dimension h = 1, while the other two have h = 0. In addition, the theory has a number of ghost fields that arise from the original R-NS description by a sequence of field redefinitions [6]. The worldsheet theory has an N = 2 superconformal symmetry [14,15], which can be enhanced to N = 4 following [16], and the physical spectrum can be characterised as a certain double cohomology of this (topological) N = 4 string [6]. (Alternatively, one may describe the physical states in terms of the critical N = 2 theory.) While this is an elegant description of the spectrum, it is actually quite complicated to determine this cohomology from first principles, and only rather partial results are known to date [17][18][19].
Just like the su(2) k algebra, also psu(1, 1|2) k has a free field realisation at level k = 1 [1,7,8], and not surprisingly this is the most efficient descripition of the theory. For example, while the localisation property of the worldsheet correlators could already be seen using the Ward identities of the sl(2, R) affine algebra [2], 1 the proof that the localised solution is in fact the only solution was only achieved using the free field description [12] of the BVW formalism. (The higher genus version of this analysis was later done in [13].) It was also noted in [12] that the free fields have a natural interpretation in terms of twistors, and this has opened the way towards the generalisation to AdS 5 [20,21].
In view of this situation it is therefore important to understand the free field realisation of the k = 1 BVW theory in detail, in particular with a view towards deriving its cohomology from first principles. On the face of it, this sounds like a rather straightforward exercise, but the situation is actually somewhat more complicated. The free field realisation of psu(1, 1|2) 1 is like the free field realisation of su(2) 1 in terms of four free fermions: it actually leads to u(1, 1|2) 1 (or u(2) 1 , for the case of su(2) 1 ), and in order to obtain psu(1, 1|2) 1 one needs to set to zero a u(1) field that is usually denoted by Z. Usually, this does not cause any significant problems, but in the present case the situation is more subtle since the construction of the N = 2 fields depends crucially, for example, on the central charge of the psu(1, 1|2) 1 factor (which is c = −2), while the free field realisation itself has c = 0. As a consequence (and as we explain in this paper), there does not seem to be any simple way of defining the N = 2 (or indeed the N = 4) fields in the free theory itself; instead the natural N = 2 fields have correction terms in their OPEs that are proportional to the u(1) field Z that needs to be set to zero in order to obtain psu(1, 1|2) -see eqs. (3.12) and (3.13) below.
Obviously, in order to define the cohomology as in [6], it is important that the OPEs are exactly obeyed; for example, the fact that the G + G + OPE is trivial implies that the corresponding zero mode G + 0 is nilpotent (and can hence play the role of one of the BRST JHEP08(2022)274 operators in the topological N = 4 string). Given that the correction term involves the Z field one may expect that it should be possible to restore the N = 2 structure if, at the same time, also the quotienting out of the Z field is implemented. Since the latter is effectively a coset construction, it can be described, in a standard BRST formalism, by introducing additional ghost fields [25]. Our setup is slightly different -the physical states are those of the N = 4 topological string, and not just some standard BRST cohomology -and so we cannot literally use the construction of [25]. However, it is possible to imitate aspects of their construction: after introducing the same set of ghost fields as in [25], we can modify the N = 2 (and N = 4) generators so that they form an actual N = 2 (and N = 4) algebra. Furthermore, we can argue that the corresponding N = 4 cohomology will implement both the physical state cohomology of [6], as well as remove the unwanted u(1) field Z. The paper is organised as follows. In section 2 we give a brief review of the BVW construction for the N = 2 algebra; this follows largely their work, but we spell out some of the details of the relevant calculations that play an important role in the following. We also discuss the construction of the global psu(1, 1|2) charges in section 2.3. In section 3 we make a natural ansatz for the N = 2 generators in terms of the free fields, and explain in detail which OPEs get modified, see eqs. (3.12) and (3.13). We also discuss the fate of the global symmetry generators in section 3.2. In section 4.1 we add the additional ghost fields and find the actual N = 2 (and N = 4) algebra generators. We also study its cohomology in section 4.2, and discuss how the global psu(1, 1|2) charges work in this modified setting. Our conclusions are summarised in section 5, and there are a number of appendices where some of our conventions and the technical details of our analysis are spelled out.

A review of the BVW construction for the N = algebra
In the hybrid formulation of Berkovits, Vafa and Witten [6] the physical states are characterised by a certain double coholomogy whose BRST operators come from an N = 4 superconformal algebra. (Alternatively, one can characterise the physical states as those of the critical N = 2 theory [14][15][16].) In this section we briefly review the construction of this N = 2 algebra -the extension to the N = 4 algebra is then relatively straightforward. Following [6] we shall begin by considering the flat space situation, and then generalise to AdS 3 × S 3 . We shall also discuss how the global supersymmetries of the string background are realised on the worldsheet.

Flat space
Let us begin by reviewing the N = 2 algebra of [6] for the case of a flat space background, i.e. for R 6 × T 4 . According to section 4 of [6], the N = 2 generators take the form as follows upon applying a similarity transformation to the generators of [14]. Here x m , and p a , θ a are the flat-space fields, with the bosonic x m fields satisfying where η mn is the 6d Euclidean metric, and m runs from 1, . . . , 6. The fermionic degrees of freedom are described in a first-order formalism, and we have where a, b ∈ {1, 2, 3, 4}, and (p, θ) have conformal dimensions (1, 0). The corresponding flat space stress tensor T flat is then given by In addition, we have the twisted N = 2 fields T C , G ± C and J C , associated to the T 4 factorwe will sometimes refer to them as the 'compact' generators -while ρ and σ are ghost fields satisfying Their central charges are c(ρ) = 28 and c(σ) = −26, and σ is the bosonisation of the usual (b, c) diffeomorphism ghosts. We should mention that relative to [6], we have changed the sign of the Q 1 term; this sign is irrelevant for the N = 2 algebra, but is required in order to ensure that the generators commute with the global symmetry generators, see below.
We have also changed some other factors and signs relative to [6] so that these generators conform with our conventions for the N = 2 algebra, see appendix A.1. Finally, all of the above expressions are suitably normal ordered, and we shall discuss the precise definition of the normal ordering in more detail for the curved background below. It is in principle straightforward to check that these generators indeed satisfy a twisted N = 2 algebra with c = 6, but the calculation is a bit tedious. 2 Since we are about to discuss the generalisation to the curved set-up (where we replace the flat-space fields by the generators of psu(1, 1|2) k ), let us highlight a few key steps and phrase them in ways that will generalise. First of all, the most non-trivial relation to check is that the G + G + OPE is regular. Since the different terms in G + come with different exponentials of the ghost fields, this is equivalent to requiring that we have separately

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The other combinations are easily seen to be regular; for example, the Q 2 Q 2 OPE is regular since the p's do not have contractions among themselves, while the contraction of the exponentials gives which, by expanding to second order in (z − w), is regular. (Here (p) 4 = p 1 p 2 p 3 p 4 .) The other terms work similarly. Returning to the above terms, we note that the Q 0 Q 0 OPE, see eq. (2.11), is regular provided that T flat has central charge c = −2, see appendix D.1 for more details. This is the case, as follows from (2.7).
Next we consider the Q 1 Q 0 + Q 0 Q 1 term, see eq. (2.10). If we write Q 1 = e −ρ R, where R does not depend on the ghost fields, one can show that eq. (2.10) is regular provided that R is a primary field of weight 3 with respect to T flat . For the situation at hand this is the case.
Unfortunately, for the Q 2 Q 0 + Q 1 Q 1 + Q 0 Q 2 term, see eq. (2.9), there is no generic simplification, and we need to work it out explicitly. This is straightforward to do in the flat space case, and the complete expression is indeed regular.

Generalisation to AdS 3 × S 3
Next we want to replace the R 6 fields by the generators of psu(1, 1|2) k . Following [6] the corresponding N = 2 generators are then where now (2.14) Here K ab are the bosonic and S a α the fermionic generators of psu(1, 1|2) k , see appendix A.2 for our conventions. All of these expressions are normal ordered with the prescription that

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and the sign depends on whether at least one of the currents A and B is bosonic, or whether both are fermionic. We should mention that this does not uniquely specify the normal ordering of the R term (since it involves more than two generators), and the correct prescription is described in eqs. (A.12) and (A.13). 3 Given our comments above, it is relatively straightforward to check that this realises indeed the N = 2 algebra. In particular, T int has central charge c = −2, and hence (2.11) is regular, while (2.10) is regular because R, defined in eq. (2.16), is indeed primary with respect to T int of conformal weight 3. We have also checked that the Q 2 Q 0 + Q 1 Q 1 + Q 0 Q 2 term in (2.9) is regular: it is easy to see that the third and second order poles are absent, using the OPEs between R(z)R(w) and T int (z)(S) 4 (w). This leaves us with the simple pole which gets a contribution from the zeroth order term in the OPEs between R(z)R(w) and T int (z)(S) 4 (w), and we have checked that the combined contribution vanishes.

Generators of the global symmetry
Since the N = 2 generators define effectively the BRST cohomology of [6], one should expect that they commute with the global psu(1, 1|2) symmetries of the background, which we shall denote byK ab andS a α in the following. The bosonic spacetime symmetriesK ab can be directly identified with the worldsheet zero modes,K ab = K ab 0 . This is essentially obvious since all the generators in (2.13a)-(2.13d) are singlets with respect to the sl(2, R) ⊕ su(2) subalgebra of psu(1, 1|2), i.e.
The situation is a bit more complicated for the fermionic generators of psu(1, 1|2). While we can takeS a 1 = (S a 1 ) 0 because the same identification does not work forS a 2 since the zero modes (S a 2 ) 0 do not directly anti-commute with G + , as was also already noted in [6]. Before we explain how to correct for this, let us mention that even the identity {(S a 1 ) 0 , G + } = 0 in (2.19) is not entirely obvious: while the OPE of S l 1 with Q 2 is regular (and hence (S a 1 ) 0 anticommutes with the corresponding terms in G + ), 4 the situation for Q 1 is somewhat tricky and hinges on the correct normal ordering prescription for the R term, see eqs. (A.12) and (A.13). Finally, the OPE of S a 1 with T int has a double pole, which however does not contribute after performing the contour integral to extract the zero mode (S a 1 ) 0 . Since the zero modes (S a 2 ) 0 do not directly anticommute with G + -there are non-trivial contributions both from the Q 2 and the Q 1 terms of G + -we need to modify them, and the correct prescription, following [6], is to takẽ The normal ordering of the (S) 4 term is unproblematic because of the anti-symmetry of the tensor and the fact that the S fields anti-commute. 4 Note that both e −ρ and e ±iσ are fermionic operators.

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The simple pole in the Q 2 · S a 2 OPE is now cancelled by that in the OPE of Q 1 with the correction term e −ρ−iσ S a 1 , while the simple pole in the Q 1 · S a 2 OPE cancels against that of the Q 0 · e −ρ−iσ S a 1 OPE. The former calculation is relatively straightforward, while the latter requires some care; we explain some of the details in appendix D.2. Note that this modified zero mode (2.20), together with the other zero modes, still obey the correct global psu(1, 1|2) commutation relations.

Free field realisation
It was shown in [1] that the k = 1 version of the AdS 3 × S 3 × T 4 hybrid string is dual to the symmetric orbifold of T 4 . Actually, the physical state condition for the hybrid string theory was not directly worked out in [1]. Instead, the equivalence of the hybrid description to the NS-R description for generic values of k [17][18][19] was used to determine what the physical state condition must amount to for the hybrid theory, and this was then extrapolated to the k = 1 case. However, given the significance of this worldsheet theory, it would be very important to work out the cohomology of the k = 1 theory directly.
The level k = 1 theory is most easily described in terms of a free field realisation of symplectic bosons and fermions [1] with (anti)-commutators where α, β ∈ {±}, and αβ is the anti-symmetric tensor with +− = − −+ = +1, see appendix B for more details. As a first step towards determining its cohomology from first principles, we therefore need to express the relevant BRST operators in this free field language. This would appear to be a relatively straightforward exercise, but there is actually one important subtlety that we need to deal with (and that will occupy us for most of this paper): the free fields do not actually realise directly psu(1, 1|2) 1 , but rather u(1, 1|2) 1 , see appendix A.3 for our conventions, and in order to obtain psu(1, 1|2) 1 one needs to gauge the u(1) field Z, see [1,12]. (The Z field describes the u(1) generator that distinguishes between the fundamental and anti-fundamental representation of u(1, 1|2), see the discussion below eq. (B.3).) For example, the central charge of the free fields equals c = 0 (as befits a u(1, 1|2) k theory), and the gauging by Z reduces the central charge to c = −2, the central charge of the psu(1, 1|2) k theory. 5 As we saw above the central charge actually plays a critical role in the above OPEs, since for example the vanishing of the terms in (2.11) hinge on T int having central charge c = −2, see the discussion below eq. (2.12) and below eq. (2.16).
To be more specific, let us denote the stress energy tensor of the free field theory by

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where : · · · : denotes the usual normal ordering of free fields. For example, in the NS sector where all fields are half-integer moded, it is simply defined by where a r , b s are any of the modes of the symplectic bosons or fermions, and the sign in the second line is '−' if both a and b are fermionic, and '+' otherwise. In the R sector where all fields are integer moded, we supplement this definition by setting Then this normal ordering agrees with the so-called conformal normal ordering • • · • • that we shall work with in appendix C, see also [22]. The reason why we consider the conformal normal ordering there is that this allows one to define the normal ordering of more than two fields in a straightforward manner, and this will be useful for our explicit computations. Actually, it is not difficult to see that the free field stress energy tensor (3.2) agrees with that of u(1, 1|2) 1 which we may take to be given in terms of the Halpern-Kiritsis construction [23] 5) where all expressions are normal ordered as in eq. (2.17), and we use the conventions of appendix A.3. This is to say, if we express the u(1, 1|2) 1 generators in terms of the free fields as in appendix B, we find One way to see this is to note that both T u(1,1|2) and T free have the same OPEs with all the currents of u(1, 1|2) 1 . Thus their difference commutes with all currents and hence must be null. We have also checked this statement explicitly, following the analysis in [22,24].

The N = 2 generators in the free field realisation
Next we observe that the first two lines in (3.5) formally look like the stress energy tensor of psu(1, 1|2) 1 , see eq. (A.11). However, this is a bit deceptive since the OPEs of the S currents in T u(1,1|2) , see eq. (B.5), differ from those in psu(1, 1|2) 1 by the Z-terms. Thus it is not directly possible to extract the psu(1, 1|2) 1 stress energy tensor from these expressions. One may nevertheless be tempted to define the 'fake psu(1, 1|2) 1 stress tensor' by The usual Sugawara construction is not applicable in this case, since the Killing form of u(1, 1|2) is zero [6]. However, one can use the Halpern-Kiritsis construction [23] and fix the remaining ambiguity by requiring that all the currents of u(1, 1|2)1 have spin one.

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but it actually does not satisfy the OPEs of a stress energy tensor, (3.8) and hence cannot be used as T int in the eqs. (2.13a) and (2.13b). Instead, we will set Note that this is effectively the coset stress energy tensor since is the stress energy tensor of the (U, V ) system. In particular, T int therefore commutes with the modes of U and V , and hence defines a stress energy tensor with c = −2, as is appropriate for T int . In particular, this then guarantees that the OPE Q 0 Q 0 is regular, because T int has the right central charge.
The other term that requires some care is the R-term in G + , see eq. (2.16). The details of this calculation are explained in appendix D.3, and it leads to This agrees with eq. (3.6) of [12], up to a factor of 2. Relative to [12] we have also reversed the roles of the two sets of fermions and symplectic bosons. With these definitions of T int , see eq. (3.9), and Q 1 , see eq. (3.11), we can then evaluate the OPEs of the generators defined in eq. (2.13). Unfortunately, they do not satisfy an N = 2 algebra any longer, since the two OPEs have correction terms proportional to : ZR :. This is not completely unexpected since the free fields only generate the u(1, 1|2) 1 algebra, and this differs from psu(1, 1|2) 1 by terms proportional to Z, see eq. (B.5). We mention in passing that these correction terms are a consequence of the fact that R defined in (3.11) is not a primary field of weight 3 w.r.t. T int = T free + ZY since it has a non-trivial OPE with Y ; more specifically we have (3.14) free field versions of eq. (2.13), and hence we still haveS a 1 = (S a 1 ) 0 .) ExpressingS a 2 in terms of the free fields we findS

The global symmetry generators
where we now label them in terms of (αβ) ∼ = a. The anti-commutator ofS ++ 2 with G + -here we use the free field version of eq. (2.13b) with T int and Q 1 defined in eqs. (3.9) and (3.11), respectively -then equals 16) where for the terms that involve more than two fields we have used the conformal normal ordering prescription of appendix C. This expression is again proportional to Z since we may write it as Thus if we quotient out by Z suitably, the generator (3.15) will commute with G + . One way to achieve this will be explained in the next section.

An N = algebra for the free-field representation
As we have seen in the previous section, it does not seem possible to construct an actual N = 2 algebra in terms of the free fields of the level k = 1 theory. The reason for this difficulty is that the free fields only realise u(1, 1|2) 1 , and that in order to reduce this to psu(1, 1|2) 1 one needs to gauge by the Z field. As a consequence, this Z field 'contaminates' the N = 2 relations. In this section we want to explain how we can recover an honest N = 2 algebra if we add additional ghost fields that incorporate the gauging of the Z field. We shall also argue that the corresponding BRST cohomology imposes then both the Z gauging condition, as well as the physical state condition of the hybrid string theory. Finally we shall show that the global psu(1, 1|2) generators commute with the N = 2 algebra (as well as the extended N = 4 algebra we are also about to discuss, see eqs. (4.5) and (4.6) below) on the corresponding physical states. This extended set-up, including the additional ghosts that we are about to introduce, is therefore a good starting point for further discussions of this worldsheet theory.

An actual N = 2 algebra
As we have explained above, the free field theory of the symplectic bosons and fermions does not quite lead to the psu(1, 1|2) 1 worldsheet theory we need, but rather to an u(1, 1|2) 1 theory. In order to reduce this to psu(1, 1|2) 1 we need to gauge the Z field, see e.g. eq. (B.5). This gauging can be performed by introducing additional ghost fields (and u(1) fields), and defining a suitable BRST operator as in [25]. Our setting is slightly different, however, since we want to describe the cohomology in terms of a topological N = 4 string, and thus the BRST operators should come from the G + supercurrents. As a consequence, we cannot literally follow the construction of [25]: if we were to modify T int by replacing the ZY term in (3.9) by Z Y , where Z and Y are the additional u(1) fields, and then simply add c(Z + Z ) to G + , where c is one of the additional ghosts, then R would be primary of weight 3 w.r.t. the new stress energy tensor, but the G + G + OPE would still not be zero, but rather equal We shall therefore follow a slightly different route in that we shall not directly use the construction of [25], but rather be inspired by their general setup. More specifically, we propose to add the following ghost fields to our set-up: • an anti-commuting ghost (b, c) with weights (1, 0) and • a commuting ghost (β , γ ) with weights (1, 0) and OPE β (z)γ (w) ∼ 1 z−w . We note that these ghosts have central charges −2, −2 and 2, respectively, and therefore their sum is indeed c = −2, as desired. Here the (b, c) and (b , c ) play the role of the ghost fields of [25], while (β , γ ) mimics the additional u(1) generators Z and Y -in the language of [25], these are associated to the H ∼ = U(1) × U(1) subgroup that is being gauged. Note that if one tried to introduce only one additional pair of (b, c) ghosts to impose the Z constraint, and then add the corresponding cZ term to G + , this would ruin the N = 2 algebra. Instead, one could try to add a term ∂(cZ) to the stress tensor, i.e. make the ansatzT int = T u(1,1|2) + T bc + ∂(cZ) in (4.2a) and (4.2b) below, but this would ruin the bosonic nature of the stress tensor. Thus one is led to introduce a pair of commuting (β , γ ) ghosts to impose the Z constraint. However, this then changes the central charge to c = 2, and one therefore needs the two (b, c) and (b , c ) ghosts to reduce it again to c = −2; this also motivates the above ghost field spectrum.
We then claim that the following generators satisfy a twisted N = 2 superconformal algebra:
Furthermore, the R field in G + is defined as in eq. (3.11). Note that the Q 2 term in G + of section 2.2, see eqs. (2.13b) and (2.15), actually vanishes in the free field realisation, as was already noted in [12]. It is relatively straightforward to show thatT int of eq. (4.3) defines a stress-tensor with central charge c = −2. Moreover, the OPE of R with Z is trivial, and therefore R is a primary of weight 3 with respect toT int -note thatT int does not involve the ZY term of (3.9) any longer. It then follows from the analysis in section 2.1 that the fields in eqs. (4.2) define a twisted N = 2 algebra; since the Q 2 term is absent in the free field realisation and the Q 1 Q 1 OPE is regular, all the terms in (2.9) are individually trivial.
We mention in passing that there are also other definitions that would have led to a consistent N = 2 algebra; for example, we could have corrected the R term (instead of adding the Z-dependent correction term to T int ). We could have also added γ Z instead to J in eq. (4.2d) and modified G + by adding −∂(e iσ γ Z).
As in [6] we can now enhance this N = 2 algebra to an N = 4 algebra by considering the currents J ±± = e ±(ρ+iσ+iH) (4.5) that extend the u(1) symmetry generated by J, to an su(2) symmetry. The commutator of the corresponding zero modes with the supercurrents (see eqs. (A.2)), then lead to the additional supercurrents 7 G + = 2e ρ+iH + e ρ+iσG+ C , (4.6a) These fields together then generate a twisted N = 4 algebra, see appendix A.1 for our conventions. (Note that the fields J ±± commute with γ Z, and thus their conformal dimension is unmodified by the γ Z correction term in T int .) As in [6] we can then use this N = 4 algebra to define the BRST cohomology that characterises the physical states. This will be discussed in more detail in the following section.

Cohomology and Z n = 0 condition
Recall from [6] that the physical states are characterised by the cohomology conditions We now propose that in our free field set-up we should do the same. This is to say, we define the N = 4 generators in terms of the free fields and the additional ghost fields as discussed in the previous section. We then claim that the physical states of our level k = 1 free field description are characterised by the cohomology of (4.7). We now want to argue that this will describe the correct physical spectrum. In particular, it should imply that the physical states are annihilated by Z n with n ≥ 0, i.e. that they are part of the psu(1, 1|2) 1 theory. As we argue below, this should then be sufficient to guarantee that the resulting cohomology will coincide with that of [6] as formulated directly in terms of the psu(1, 1|2) k WZW model.
Unfortunately, a complete BRST analysis of (4.7) is quite difficult (although, in this free field set-up, this is now maybe within reach); note that also the cohomology of [6] has only been very partially studied [17][18][19], and that a complete description of the physical spectrum in this language is still missing. In order to make progress we shall assume that we can choose a representative in each cohomology class that satisfies 8 b n ψ = b n ψ = β n ψ = 0 for n ≥ 0 , c n ψ = c n ψ = γ n ψ = 0 for n ≥ 1 . This seems reasonable since these conditions only refer to the additional ghosts we have added to our description. We now want to show that provided this is the case, i.e. provided that ψ is in the BRST cohomology of eq. (4.7) and satisfies eq. (4.8), then ψ also satisfies where are the 'old' N = 2 generators without the additional ghosts and the (γ Z) correction term inT int . In particular, this should mean that they are part of the original cohomology of [6]. In order to show eq. (4.9), we note that eq. (4.8) in particular means that Thus the BRST condition T 0 ψ = 0, see eq. (4.7), implies that

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where T is given in eq. (4.10a), and we have used eq. (4.11) with n = 0 for each of the additional ghost terms. We can write out the correction term in eq. (4.12) as Applying β n with n ≥ 0 to eq. (4.12), and using that β n commutes with T 0 as well as eq. (4.8), we therefore deduce that Z n ψ = 0 n ≥ 0 , (4.14) thereby giving the last identity in eq. (4.9). In fact, also the second identity in eq. (4.9) is now manifest since the correction term in eq. (4.12) vanishes, and hence It remains to show that ψ is also annihilated by G + 0 . Using that the full G + 0 annihilates ψ, see eq. (4.7), we have where c = e iσ is the c-ghost of the original hybrid formulation 9 and Here we have used that eqs. (4.11) and (4.14) imply 18) and hence that only the negative modes less than −1 of T extra enter in eq. (4.16). Furthermore, using the gauge equivalence in eq. (4.7) we should be able to find a representative ψ for which c n ψ = 0 n ≥ 1 . Thus the remaining terms in eq. (4.16) vanish, and we conclude that we can always find a representative for which G + 0 ψ = 0. This completes our argument.

Global symmetry
Finally, we need to address the question of whether the global symmetries in psu(1, 1|2) commmute with the N = 2 algebra, see the discussion in sections 2.3 and 3.2. As was explained in section 2.3, one would naively expect the zero modes of psu(1, 1|2) k to commute with the N = 2 generators. While this is true on the nose for all the bosonic and half of the fermionic generators, we had to modify half of the fermionic generators so as to make them (anti)-commute, see eq. (2.20). Furthermore, the construction broke down again in the free field realisation since there were Z dependent correction terms, see eq. (3.17).
Here we will explain that while these psu(1, 1|2) generators do not (anti)-commute with

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the N = 4 generators in the free-field realisation (i.e. the generators in eqs. (4.2), eq. (4.5) and eqs. (4.6)), these (anti)-commutators vanish on physical states, and hence the physical states sit in representations of psu(1, 1|2). Let us start by noting that our N = 2 generators only differ from those in section 3.2 by the fact that we have modified the stress energy tensor, i.e. replaced (4.20) Since all the generators of psu(1, 1|2) 1 commute with Z n (and do not involve any of the additional ghost fields), the only interesting correction term is the −ZY term. It has an effect since the fermionic generators do not commute with Y n , As a consequence, the right-hand-side of (3.17) will still only involve terms proportional to Z, and they are removed by the gauge condition Z n ψ = 0 for n ≥ 0, see eq. (4.9), resp. are null because Z † m = −Z −m . Finally, we note that the global symmetry generators also commute with the J ±± generators of eq. (4.5) since the latter do not involve any psu(1, 1|2) 1 generators, and the ghost factor e −ρ−iσ in eq. (3.15) has a regular OPE with J ±± . Thus, the global symmetry generators also commute on physical states with the N = 4 algebra generated by these fields, and this is sufficient to deduce that the physical states must sit in representations of the global symmetry algebra psu(1, 1|2), as expected.

Conclusion
In this paper we have undertaken first steps towards writing the BVW hybrid worldsheet theory [6] for the pure k = 1 NS-NS background in terms of free fields, using the free field realisation of the psu(1, 1|2) 1 affine algebra. As we stressed throughout, the main subtlety with this construction is related to the fact that the free fields only realise u(1, 1|2) 1 , and that one needs to gauge the u(1) Z-field in order to obtain actually psu(1, 1|2) 1 . In particular, this is responsible for contaminating the N = 2 algebra relations that underpin the BVW construction in the (naive) free field realisation.
As we explained in section 4 it is possible to overcome these difficulties by introducing additional ghost systems that implement the Z gauging, following at least in spirit the construction of [25]. In particular, this allowed us to construct an N = 2 algebra in our free JHEP08(2022)274 field setting, and it seems plausible that the cohomology of the associated N = 4 topological string will lead to the correct physical spectrum, see section 4.2.
There are a number of natural future directions. First of all, it would be very important to work out the cohomology of our N = 4 system of section 4 from first principles; given that this is now a free field construction, this should be feasible, and it would be very satisfying if this could be shown to reproduce what is expected from the arguments in [1]. In turn, this should also allow us to work out the correlators of [12] more fully -in particular, it should then be possible to determine the actual ghost contributions and hence check whether the undetermined coefficients in [12] reproduce what is expected from the dual symmetric orbifold, see e.g. [26].
The other natural direction is to try and generalise these considerations to the worldsheet theory that has been proposed to be exactly dual to free SYM in 4D [20,21]. This worldsheet theory essentially consists of twice the symplectic bosons and fermions, but also needs to be quotiented out by the analogue of the Z-field -this was denoted by C in [20,21]. One should therefore be able to construct an N = 4 superconformal algebra for this theory, following essentially the same arguments as in this paper, and it would be interesting to see whether this can be further extended to N = 8.

A Conventions
In this appendix we collect various conventions.

A.1 The N = 2 algebra and the topological twist
Let us start by reviewing the topologically twisted N = 2 superconformal algebra with central charge c and introduce our conventions. The fields of this algebra consist of the stress-tensor T with modes L n , a u(1) current J with modes J n , and two supercurrents G ± with modes G ± n . Here we consider the usual topological twist where we add 1 2 ∂J to the stress-tensor T [27]. Then the resulting stress tensor has vanishing central charge, while the conformal dimensions of the primary fields are shifted according to their u(1) charge, e.g. G + and G − will have weights 1 and 2, respectively. The resulting topologically twisted algebra has then the commutation relations We shall also need the extension of this algebra to the topologically twisted N = 4 algebra; the latter has, in addition, the generators J ++ m , J −− m andG ± m , with the non-trivial commutation relations being The other important algebra that will play a significant role in our analysis is psu(1, 1|2) k . There are two natural conventions in which one may describe it, namely those of [6], and those of [1,12]. We shall also spell out how these conventions are related to one another.
In the conventions of [6], psu(1, 1|2) k consists of the bosonic generators K ab m = −K ba m that define an so(2|2) 1 algebra, and the two fermionic generators (S a α ) m , α = 1, 2, each of which sits in the vector representation of this so(2|2) algebra. (Here the latin indices run over a ∈ {1, 2, 3, 4} and label this vector representation.) The commutation relations are K ab m , K cd n = mk abcd δ m+n,0 + δ ac K bd m+n − δ ad K bc m+n − δ bc K ad m+n + δ bd K ac m+n , where the tensors are totally anti-symmetric with the convention that 12 = 1 = 1234 .

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On the other hand, in the conventions of [1,12], the bosonic generators of psu(1, 1|2) k are J a m ∈ sl(2, R) k and K a m ∈ su(2) k , where now a ∈ {3, ±}. The fermionic generators, on the other hand, are denoted by S αβγ m , where now α, β, γ ∈ {±}, and α and β are spinor indices with respect to sl(2, R) and su(2), respectively, while γ accounts for the fact that there are two such bi-spinorsγ therefore plays the role of α in S a α , see also footnote 11 below.
In these conventions the commutation relations of psu(1, 1|2) k are where the σ a are Pauli matrices and c a = ±1, see appendix A of [1] for more details.
The two descriptions 11 are equivalent to one another, see also [19], and the translation between the different conventions is while for the fermions we have (A.10) 11 The last index can be raised and lowered via √ 2S αβ+ = S αβ 2 and − √ 2S αβ− = S αβ 1 .

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Finally, the stress tensor of psu(1, 1|2) k equals in either convention (A.11) where : · · · : denotes the normal ordering defined in eq. (2.17). Finally, we normal order the generators in (2.16) as along with the OPEs of SS that have a Z term, Note that the difference to the psu(1, 1|2) 1 current OPEs are precisely these Z-terms, and thus we need to 'gauge' by the Z field in order to reduce u(1, 1|2) 1 to psu(1, 1|2) 1 . More structurally, the fact that these free fields do not directly lead to psu(1, 1|2) 1 is a reflection of the fact that they only sit in a representation of u(1, 1|2), namely the fundamental resp. the anti-fundamental, but not in a representation of psu(1, 1|2).

C Normal ordering conventions
Normal ordering for products of two fields is relatively straightforward, see e.g. eq. (2.17), but whenever more fields are involved, as is for example the case in the definition of (2.15) and (2.16), there are ambiguities, and it will be important for us to fix them. In the following we will explain, following [22], a definite prescription for how to normal order an arbitrary number of fields. We will denote the corresponding normal ordering by • • · • • and sometimes refer to it as 'conformal normal ordering'. For the free symplectic bosons and fermions fields, this normal ordering agrees with (3.3) and (3.4) when restricted to two fields. However, we will also apply it in general, and this is how we define the triple products in (2.15) and (2.16).
In the following we shall first explain the prescription for a free boson field; at the end of this appendix we shall also explain how it works for the bc system, as well as a chiral boson, see eqs. (C.13) and (C.15) below. The OPE of the free boson field X(z) is We define the normal ordering by where the first term on the right hand side is radially ordered, i.e. the field with bigger modulus stands to the left of the one with smaller modulus. Our normal ordering is therefore equivalent to acting by a differential operator We can invert this relation to give

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since the differential operators in the exponent commute with one another. The normal ordering of three or more fields can then be defined similarly, i.e.
(C.5) where the fields on the right hand side are again radially ordered. We can also invert this relation as in eq. (C.4).
It should now be clear how this can also be generalised to obtain the radial ordering of two normal ordered operators where F , G are functions of the field X. Using the above formula for normal ordering,we have , (C.8) and δ δ F X(x) is the differentiation with respect to X(x) of F (X), and similarly for G. Altogether, this therefore leads to (C.9) We can, in particular, use this formula to compute radial orderings of products of exponentials. Since exp(imX(z)) exp(inX(w)) is an eigenfunction of i.e., eq. (C.9) becomes

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The generalisation from the free boson to a (b, c) system is straightforward. For a (b, c) system we have instead of (C.1), and hence (C.14) Finally, for a chiral boson with OPE

D Checking the N = 2 algebra
In this appendix we give some details concerning the calculation of the OPEs of the N = 2 generators of eq. (2.13).

D.1 Calculation of the G + G + OPE
Let us start with the G + G + OPE. We shall concentrate on the computation of the Q 0 Q 0 OPE, which contains the contribution, see eq. (2.13b) -the remaining terms will be discussed below (D.1) Here we are using the conformal normal ordering convention, which was explained in appendix C. By the usual chain rule argument we see that , .

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The contraction of the ghost terms can be done using the same methods as in appendix D.1, and it leads to a factor of On the other hand, the contribution coming from the OPE of S l 1 (z) with T int (w) gives rise to (D.12) Expanding out the exponential in (D.12), and concentrating on the simple pole -this is the only contribution that survives after taking the contour integral -the first term in the bracket leads to contributions that cancel exactly the simple pole in (D.11). On the other hand, the second term in the bracket of (D.12) cancels against part of the simple pole of S l 2 (z)Q 1 (w), that can be calculated directly. The terms that remain from the simple pole ofS l 2 (z)G + (w) are proportional to e −ρ Φ, 12 where , (D.13) and we have used that lamn K mn −2 = labd K cb −1 K cd −1 , (D.14) as follows from the commutation relation (A.3). By contracting the last two terms in (D.13) with lijk , we find (D. 15) By direct evaluation we see that the terms that contain (S g 1 ) −1 where g is either g = i, or g = j, or g = k vanish, and hence g has to take the remaining value g ∈ {1, 2, 3, 4}\{i, j, k}. Thus, eq. (D.15) becomes (D.17) 12 Recall that e −ρ has conformal dimension h = −2.

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where l is a free label. Contracting with pijk we get where now the free index is p. Finally, we can rewrite for fixed p the product of the two K operators on the right as (Here the factor of 1 4 arises because p is not summed over the 4 values, and the minus sign arises because we exchanged the order of the indices in the totally anti-symmetric tensor.) Then, combining (D.18) and (D. 19), it follows that Φ in eq. (D.13) indeed vanishes.

D.3 Calculation of the R in terms of the free fields
In this appendix we explain the calculation of R, see eq. (2.16) for k = 1, in terms of the free fields, see eq. (3.11). In terms of psu(1, 1|2) 1 generators the state corresponding to R equals in our conventions where we have already used that (S αβ 1 ) −1 (S γβ 1 ) −1 |0 = 0 in the free field realisation. Plugging in the free field realisation for these generators, we find for the terms in the first line

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Combining all these terms that involve bosonic generators, i.e. the K ab S a 1 S b 1 term in eq. (2.16), we thus obtain On the other hand, the 4S a 1 ∂S a 1 term eq. (2.16), i.e. the 4S∂S term in (D.20), equals Altogether we therefore find for R at k = 1, see eq. (2.16)